International Journal of Mathematics and Mathematical Sciences

Volume 2008, Article ID 597408, 8 pages

http://dx.doi.org/10.1155/2008/597408

## Bipartite Toughness and -Factors in Bipartite Graphs

^{1}School of Mathematics and System Sciences, Shandong University, Jinan 250100, China^{2}Department of Computer Science, Memorial University of Newfoundland, St. John's, NL, Canada A1B 3X5^{3}School of Computing, The University of Southern Mississippi, Hattiesburg, MS 39406, USA^{4}Department of Mathematics, College of Arts and Sciences, The State University of West Georgia, Carrollton, GA 30118, USA

Received 18 August 2008; Accepted 7 October 2008

Academic Editor: Siamak Yassemi

Copyright © 2008 Guizhen Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We define a new invariant in bipartite graphs that is analogous to the toughness and we give sufficient conditions in term of for the existence of -factors in bipartite graphs. We also show that these results are sharp.

#### 1. Introduction

*Toughness*,
like connectivity, is an important invariant in graphs. There has been extensive work on toughness (see the survey in [1]) since Chvátal introduced
the concept in 1973 [2]. The toughness of a graph is the minimum value of ,
where is a proper subset of the vertices of and is the number of connected components after
removing from .
(If is a complete graph so that is always equal to 1, then is set to be .)
That is, for any integer , cannot be split into connected components by removing less than vertices. We also say that is -tough. Chvátal made a number of conjectures
in [2], including the
famous 2-tough conjecture saying that every 2-tough graph has a Hamiltonian
cycle. Having inspired many interesting results, the 2-tough conjecture itself
was showed to be false by Bauer et al. in 2000 [3].

A subgraph of is called a *factor* of if is a spanning subgraph of .
An important class of factors is -*factors*, also called *regular degree
factors*, where every vertex of has degree in .
(Note that a perfect matching is a 1-factor, and a Hamiltonian cycle is a
connected 2-factor.) There has been extensive work on the conditions of
existence of various factors in graphs. Many results can be found in the latest
survey by Plummer [4].

It is natural to expect that toughness, yet another
measure of the connectivity of a graph, ought to relate to the existence of -factors in graphs. Enomoto et al. [5–7] proved that every -tough graph contains a -factor if it satisfies trivial necessary
conditions, and there are -tough graphs for any that do not contain a -factor. Consider a *bipartite graph * where is a partition of and is the edge set of with each edge having one end in and the other in . Katerinis [8] proved
that every 1-tough bipartite graph has a 2-factor. Recall that the toughness of
a bipartite graph is at most 1 because the removal of from (assuming ) results in an independent set .
Therefore, it is not possible to use toughness to predict the existence of -factors in balanced bipartite graphs for any .

##### 1.1. Bipartite Toughness

In this paper,
we introduce *bipartite toughness*, which is analogous to the concept of
toughness but reflects the bipartition of .
The bipartite toughness of a bipartite graph is the minimum value of ,
where is a proper subset of or and is the number of connected components after
removing from .
We set for complete bipartite graphs, just like for complete graphs.

A bipartite graph can have a regular degree factor
only if .
Therefore, in the rest of the paper, we consider only a *balanced bipartite
graph* with .
For a subset of ,
we use to denote the set of vertices adjacent to at
least one vertex in .
For two disjoint subsets and of ,
we use to stand for the number of edges having one end
in and the other in .
Other terminologies and notations used in this paper follow [9] and other references.

Bipartite toughness measures the connectivity of a bipartite graph better than toughness does. In contrast to toughness that is at most 1 in a bipartite graph, can be arbitrarily big. For example, in a complete bipartite graph with one edge deleted, which approaches to , is just like in a complete graph with one edge deleted. Interestingly, a better invariant to predict the existence of -factors in balanced bipartite graphs, for any . Furthermore, by their definitions, calculating in a bipartite graph is easier than calculating since one is a subtask of the other.

##### 1.2. Our Results

Let be a balanced bipartite graph with and be an integer. In this paper, we prove the following three theorems.

Theorem 1.1. *Let . If ,
then has a 1-factor.*

Theorem 1.2. *For and ,
if ,
then has a -factor.*

Theorem 1.3. *For ,
if ,
then has a -factor.*

These theorems together give a sharp bound of for to have a -factor, for . (See Figure 1. Note that when and is odd; and when .)

The bound of is sharp in the following senses.

(a)For Theorem 1.1, let and construct a balanced bipartite graph as follows. Let and , where , , and . Let be comprised of all possible edges between and and all possible edges between and . If is even, then we add into an edge between and . Here, so that by Lemma 2.1 below, has no 1-factor. On the other hand, it is not hard to verify that in this construction of . Therefore, is a sharp bound.(b)For Theorem 1.2, for integers and , construct a balanced bipartite graph as follows. Let and , where , , and . Let be comprised of all possible edges between and , all possible edges between and , and a 1-factor between and . Here, so that by Lemma 2.1 below, has no -factor. On the other hand, it is not hard to verify that in . Therefore, is a sharp bound.(c)For Theorem 1.3. Let and be an integer. Obviously, . Construct a balanced bipartite graph as follows. Let and where , , and . Let be comprised of all possible edges between and , all possible edges between and , and a -factor between and . Then . Again, by Lemma 2.1 below, has no -factor. Moreover, it is not hard to verify that . Therefore, is also a sharp bound.

It is also worth to mention that, unlike Enomoto et al.'s well-known result that -tough graphs have -factors, in our results the bound of is much smaller than , in fact less than 2 for most (see Figure 1). This looks counterintuitive but it is due to a (not so good) feature of . Although can approach to , most time it does not increase significantly with edge connectivity or minimum degree. For example, if , has minimum degree (say on vertex ), then removing all vertices in except would split into components. So even when is as high as .

#### 2. Proofs of the Theorems

The following lemma will be needed in the proofs of theorems.

Lemma 2.1. *Let be a balanced bipartite graph, where ,
and let be an integer. Then the following three
statements are equivalent: *

(i)* has a -factor;*(ii)* has edge-disjoint 1-factors;*(iii)*for any and , **Proof. *(i) and (ii): following the
König-Hall theorem [9, Theorem 5.2 and Lemma 5.2], a regular degree bipartite graph has a perfect
matching. Therefore, a -factor of a bipartite graph can be partitioned into a collection of edge-disjoint perfect matchings (1-factors).
(ii) to (i) is trivial.

(i) and (iii): the equivalence of (i) and (iii) can be
deduced from the max-flow min-cut theorem [10, 11]. Convert into a network by (a) adding a source vertex with multiedges between and each vertex ;
(b) adding a sink vertex with multiedges between and each vertex ;
and (c) orienting each edge into a directed arc going from to ,
from to ,
or from to (see Figure 2). Clearly, has a -factor the network has a -flow from to any cut in the network that separates and contains at least forward edges. For any and ,
consider the cut shown in dashed line in Figure 2, we haveso that

*Proof. *Suppose has no -factor and ,
we will infer that .
According to Lemma 2.1, there exist and such that .
Let and .
ThenObviously, .
We can further assume thatBecause, if ,
then we can let and and have , ,
and .
By symmetry, this converts to the case of .

We then have two cases to consider.
*Case 1. *

If ,
then by (2.3). By and (2.4), we have ,
where . ThusThis completes the proof of
Theorem 1.1. (Note that when ,
we have only Case 1 to consider.)

*Proof. *Now suppose ,
by (2.5), we have .
Let .
Then by (2.3), .
Let .
Then and .
Therefore,*Case 2. *If ,
then we have .
By (2.4) and (2.7),*Case 2. *If ,
then we have .
By (2.5) and (2.7),*Case 2. *

Let be the unique integer satisfyingBy (2.10), .
By (2.3) and (2.11), there is a vertex that is adjacent to at most vertices in .
Let so and .
By (2.4) and (2.11), we have .
Therefore,Define a function .
It is easy to verify that, by the assumption of , .
Since is a convex function, it follows that for .
By (2.12),

This completes the proof of
Theorem 1.2.

*Proof. *Indeed, we will prove that the result in Theorem 1.3
holds for all .
The condition of in Theorem 1.3 is only because that is not as tight a bound as when .

Suppose has no -factor, we will infer that .
According to Lemma 2.1, there exist and such thatwhere and .
Like in the proof of Theorems 1.1 and 1.2, we can still assume (2.4).

Suppose is vertex in that is adjacent to the least number (denoted
by ) of vertices in .
By (2.14), we have .
Then with (2.4), we further have .
Let ,
then and .
Therefore,

This completes
the proof of Theorem 1.3.

#### 3. Conclusion and Future Work

We have defined a new invariant in bipartite graphs called bipartite toughness and provided a sharp bound of it for a balanced bipartite graph to have a -factor, for from 1 through . We view this as a big improvement from using toughness to predict -factors in bipartite graphs, as toughness of a bipartite graph is at most 1 and it cannot predict -factors for any .

There is also research on computational complexity of
toughness. In general, recognizing toughness of a graph is NP-hard [12]. Furthermore, 1-tough of
graphs is also NP-hard [13],
and even 1-tough of bipartite graphs is NP-hard [14] too. Toughness in claw-free
(-free) graphs [15], 1-tough in split graphs
[14], and toughness in
split graphs [16] have
been shown in *P*. In the future, it would be very interesting to determine the
complexity of bipartite toughness.

#### Acknowledgment

The first author's work is partially supported by National Natural Science Foundation of China NSFC 10871119.

#### References

- D. Bauer, H. Broersma, and E. Schmeichel, “Toughness in graphs—a survey,”
*Graphs and Combinatorics*, vol. 22, no. 1, pp. 1–35, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Chvátal, “Tough graphs and Hamiltonian circuits,”
*Discrete Mathematics*, vol. 5, no. 3, pp. 215–228, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Bauer, H. J. Broersma, and H. J. Veldman, “Not every 2-tough graph is Hamiltonian,”
*Discrete Applied Mathematics*, vol. 99, no. 1–3, pp. 317–321, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. D. Plummer, “Graph factors and factorization: 1985–2003: a survey,”
*Discrete Mathematics*, vol. 307, no. 7-8, pp. 791–821, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Enomoto, B. Jackson, P. Katerinis, and A. Saito, “Toughness and the existence of $k$-factors,”
*Journal of Graph Theory*, vol. 9, no. 1, pp. 87–95, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Enomoto, “Toughness and the existence of $k$-factors. II,”
*Graphs and Combinatorics*, vol. 2, no. 1, pp. 37–42, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Enomoto, “Toughness and the existence of $k$-factors. III,”
*Discrete Mathematics*, vol. 189, no. 1–3, pp. 277–282, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Katerinis, “Two sufficient conditions for a 2-factor in a bipartite graph,”
*Journal of Graph Theory*, vol. 11, no. 1, pp. 1–6, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Bondy and U. S. R. Murty,
*Graph Theory with Applications*, Macmillan, New York, NY, USA, 1976. View at MathSciNet - J. Folkman and D. R. Fulkerson, “Flows in infinite graphs,”
*Journal of Combinatorial Theory*, vol. 8, pp. 30–44, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Ore,
*Theory of Graphs*, American Mathematical Society, Providence, RI, USA, 1962. View at Zentralblatt MATH · View at MathSciNet - D. Bauer, S. L. Hakimi, and E. Schmeichel, “Recognizing tough graphs is NP-hard,”
*Discrete Applied Mathematics*, vol. 28, no. 3, pp. 191–195, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Bauer, E. Schmeichel, and H. J. Veldman, “Some recent results on long cycles in tough graphs,” in
*Proceedings of the 6th Quadrennial International Conference on the Theory and Applications of Graphs*, pp. 113–123, Kalamazoo, Mich, USA, May-June 1991. View at Zentralblatt MATH · View at MathSciNet - D. Kratsch, J. Lehel, and H. Müller, “Toughness, hamiltonicity and split graphs,”
*Discrete Mathematics*, vol. 150, no. 1–3, pp. 231–245, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Matthews and D. P. Sumner, “Hamiltonian results in ${K}_{1,3}$-free graphs,”
*Journal of Graph Theory*, vol. 8, no. 1, pp. 139–146, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. J. Woeginger, “The toughness of split graphs,”
*Discrete Mathematics*, vol. 190, no. 1–3, pp. 295–297, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet